Real Gas Radiation¶
This notebook describes the real gas radiation model and goes through some examples of things that can be done with it.
Model Description¶
Flux equations¶
The difference between this model and the grey gas model is that now the optical depth depends on frequency (or wavenumber $\nu$ here). To derive the model flux equations in this case, we start with equations $(17)$ and $(18)$ from the grey_radiation notebook:
\begin{equation}
\frac{dF^{\uparrow}_{\nu}}{d\tau_{\nu}} = F^{\uparrow}_{\nu} - \pi B(\nu, T)
\label{eq: 1}\tag{1}
\end{equation}\begin{equation}
\frac{dF^{\downarrow}_{\nu}}{d\tau_{\nu}} = -F^{\downarrow}_{\nu} + \pi B(\nu, T)
\label{eq: 2}\tag{2}
\end{equation}
where:
\begin{equation}
d\tau_{\nu} = -\rho kdz = \frac{k}{g}dp
\label{eq: 3}\tag{3}
\end{equation}
using the hydrostatic equation $\partial p/\partial z = - g\rho$. $k = \sum_{i=0}^n k_iq_i$ where $q_i(z)$ is the specific humidity ($q_i = \rho_i / \rho$). $k_i$ is an absorption coefficient and has units of area per unit mass ($m^2kg^{-1}$). $i$ refers to a particular molecule, $k_i$ depends on wavenumber $\nu$ and pressure $p$ whereas $q_i$ just depends on pressure. Note $\tau_{\nu}$ increases towards the surface and is $0$ at the top of the atmosphere where $p=0$. $B(\nu, T)$ is the blackbody distribution.
By combining the homogenous and particular solutions, the solution to $(1)$ is:
\begin{equation}
F^{\uparrow}_{\nu}(\tau_{\nu}) = F^{\uparrow}_{\nu}(\tau_{\nu}^s)e^{\tau_{\nu} - \tau_{\nu}^s} +
\int_{\tau_{\nu}}^{\tau_{\nu}^s} \pi B(\nu, T) e^{\tau_{\nu} - \tau_{\nu}'} d\tau_{\nu}'
\label{eq: 4}\tag{4}
\end{equation}
where $\tau_{\nu}^s$ is the optical depth at the surface for wavenumber $\nu$ and we assume we know the surface flux value $F^{\uparrow}_{\nu}(\tau_{\nu}^s)$, i.e. blackbody spectrum of planet surface.
If we then define the transmission function over a wavenumber interval $\Delta \nu$ as a function of two pressures $p_1$ and $p_2$ to be:
\begin{equation}
\mathcal{T}_{\Delta \nu}(p_1, p_2) = \frac{1}{\Delta \nu} \int_{\Delta \nu} e^{\tau_{\nu}(p_1) - \tau_{\nu}(p_2)} d\nu
\label{eq: 5}\tag{5}
\end{equation}
We can then integrate $(4)$ over wavenumber and convert the $\tau$ integral to a $p$ integral to give:
\begin{equation}
F^{\uparrow}_{\Delta \nu}(p) = \frac{1}{\Delta \nu} \int_{\Delta \nu} F^{\uparrow}_{\nu}(p) = F^{\uparrow}_{\nu}(p_s)\mathcal{T}_{\Delta \nu}(p, p_s) - \int_{p}^{p_s} \pi B(\nu, T(p')) \frac{d\mathcal{T}_{\Delta \nu}(p, p')}{dp'} dp'
\label{eq: 6}\tag{6}
\end{equation}
Doing the same for the downward flux, assuming we know the top of atmosphere flux value $F^{\downarrow}_{\nu}(0)$, i.e. blackbody spectrum of star, we get:
\begin{equation}
F^{\downarrow}_{\Delta \nu}(p) = F^{\downarrow}_{\nu}(0)\mathcal{T}_{\Delta \nu}(0, p) +
\int_{0}^{p} \pi B(\nu, T(p')) \frac{d\mathcal{T}_{\Delta \nu}(p', p)}{dp'} dp'
\label{eq: 7}\tag{7}
\end{equation}
Summing over the wavenumber bands, we get the net flux:
\begin{equation}
F(p) = \sum_{\Delta \nu} \left(F^{\uparrow}_{\Delta \nu}(p) - F^{\downarrow}_{\Delta \nu}(p)\right)\Delta \nu
\label{eq: 8}\tag{8}
\end{equation}
We can then pass this flux into equation $(11)$ of the grey_radiation notebook to compute the evolution of temperature with time. Once the flux has been computed, the rest of the simulation is done in exactly the same way as in the grey gas case.
